Question

If n is an odd integer, then show that n² – 1 is divisible by 8.

Answer 1

Solution:
Any odd integer n is of the form 4m + 1 or 4m + 3.

=> n² – 1 = (4m + 1)² – 1
= 16m² + 8m
= 8(2m² + m),
which is divisible by 8.
Also, n² – 1 = (4m + 3)² – 1
= 16m² + 24m + 8
= 8(2m² + 3m + 1),
which is divisible by 8.
Hence, n² – 1 is divisible by 8 for any odd integer n.